A paradox in game theory, has been described as:A combination of losing strategies becomes a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996. A more explanatory description is:There exist pairs of games, each with a higher probability of losing than winning, for which it is possible to construct a winning strategy by playing the games alternately.

A simple example of how and why the paradox works, again consider two games Game A and Game B, this time with the following rules:In Game A, you simply lose $1 every time you play.In Game B, you count how much money you have left. If it is an even number, you win $3. Otherwise you lose $5.Say you begin with $100 in your pocket. If you start playing Game A exclusively, you will obviously lose all your money in 100 rounds. Similarly, if you decide to play Game B exclusively, you will also lose all your money in 100 rounds.However, consider playing the games alternatively, starting with Game B, followed by A, then by B, and so on (BABABA...). It should be easy to see that you will steadily earn a total of $2 for every two games.Thus, even though each game is a losing proposition if played alone, because the results of Game B are affected by Game A, the sequence in which the games are played can affect how often Game B earns you money, and subsequently the result is different from the case where either game is played by itself.

Don't want to rain on your parade - and perhaps some better minds than mine might like to chip in - but this is just a variation of the well-known mathematical ratchet theory. It doesn't work in gambling because it requires at least one game to have a certain profit (or to put it another way, a positive EV) and, as we all know, there is no such thing in a gambling house.

Don't want to rain on your parade - and perhaps some better minds than mine might like to chip in - but this is just a variation of the well-known mathematical ratchet theory. It doesn't work in gambling because it requires at least one game to have a certain profit (or to put it another way, a positive EV) and, as we all know, there is no such thing in a gambling house.

Perhaps you've missed something: "even though each game is a losing proposition if played alone, because the results of Game B are affected by Game A, the sequence in which the games are played can affect how often Game B earns you money, and subsequently the result is different from the case where either game is played by itself."

(The bolding in the following is mine)While Game B is a losing game under the probability distribution that results for modulo when it is played individually (modulo is the remainder when is divided by ), it can be a winning game under other distributions, as there is at least one state in which its expectation is positive.

"a dependence between Games A and B, so that a player is more likely to enter states in which Game B has a positive expectation, allowing it to overcome the losses from Game A

and

"it is actually a combination of three simple games: two of which have losing probabilities and one of which has a high probability of winning."

You have to be able to CHOOSE a game in which you WILL win $3. This is a positive EV game and no such game exists in gambling houses.

Note - what appears in my message above looks NOTHING LIKE that which I created. All the formatting has been completed crapped up! However, I guess the point still gets across. Why can't forum writers write decent s/w??

Still you don't get it,don't you?Game A and game B are 2 different options,but in order their outcomes to be dependable,we need to connect them (Game A and B) in a pair,therefore 2 different games to be played as ONE game.Replace Game A with ODD and game B with EVEN and instantly you understand the correlation in the decisions.Was that so hard to understand?

In Game B, we first determine if our capital is a multiple of some integer . If it is, we toss a biased coin, Coin 2, with probability of winning . If it is not, we toss another biased coin, Coin 3, with probability of winning . The role of modulo provides the periodicity as in the ratchet teeth.

The probability of coin 3 winning is around 0.75 - the coin is biased, and so the expectation is therefore POSITIVE. Someone asked about Parrondo's paradox at the Wizard of Odds site:http://wizardofodds.com/ask-the-wizard/149/

Actually, a better explanation of why PP can't work with casino games is because outcomes are independent, but PP requires some interaction between the current game and the previous one. Taken both together, the games do result in an overall negative expectation, but the crucial part is being able to select the game which has a positive expectation given what's just happened. But since what just happened has no effect on expectation in casino games, PP cannot work with them.

Actually, a better explanation of why PP can't work with casino games is because outcomes are independent, but PP requires some interaction between the current game and the previous one. Taken both together, the games do result in an overall negative expectation, but the crucial part is being able to select the game which has a positive expectation given what's just happened. But since what just happened has no effect on expectation in casino games, PP cannot work with them.

Granted, this isn't easy to see at first glance.

Still you don't get it,don't you?Game A and game B are 2 different options,but in order their outcomes to be dependable,we need to connect them (Game A and B) in a pair,therefore 2 different games to be played as ONE game.Replace Game A with ODD and game B with EVEN and instantly you understand the correlation in the decisions.

Game B must have 2 variants that are choosable between. The first variant has a positive EV and the second variant the usual negative EV. Read and try to understand the link I provided. But - whatever.

Blue Angel I do not understand this topid.Please give a desription of the game A and the game B. Chance the dollars in units. The time we put money on the table is long ago .

For example let game A be LOW and game B the HIGH.If one of the 2 options loses it's because the other wins,therefore a sensible strategy would be to emphasize the wins and minimize the losses.This possible under certain circumstances and criteria applied.In the short run almost always there is unbalance because of the variance,so why play against it and not with it?!